bibkey: schulte2015a077864 authors: Werner Schulte year: 2015 title: “OEIS A077864, Expansion of (1-x)^(-1)/(1-x-2x^2-x^3)” doi: null url: https://oeis.org/A077864 claim: “%N Expansion of (1-x)^(-1)/(1-x-2x^2-x^3). %F a(0)=1, a(1)=2, a(2)=5, a(3)=11, a(n)=2a(n-1)+a(n-2)-a(n-3)-a(n-4) for n>3. - Philippe Deléham, Oct 25 2006 %F Conjecture: a(n) = Sum_{j=0..n/2} A027907(n+1-j,2j+1), n >= 0. - Werner Schulte, Sep 29 2015 %N Triangle of trinomial coefficients T(n,k) (n >= 0, 0 <= k <= 2*n), read by rows: n-th row is obtained by expanding (1 + x + x^2)^n.” strata_touched:

• D5/S1/Recurrence/Invariants/TrinomialOddDiagonalRationalSeries license: citation-only triage: anchor

OEIS A077864

The OEIS entry records the expansion (1-x)^(-1)/(1-x-2*x^2-x^3) and the initial values 1, 2, 5, 11, .... Philippe Deléham’s formula gives the order-four recurrence with these initial values. Werner Schulte’s 2015 formula conjectures that the coefficient at n is the odd diagonal sum of the trinomial triangle A027907.

A027907 defines T(n,k) by the coefficient of x^k in (1+x+x^2)^n, read row by row. The Lean definitions use the same polynomial coefficient and the same rational generating series.

Verified locator

• URL: https://oeis.org/A077864
• Locator: FORMULA, Deléham, Oct 25 2006; FORMULA, Werner Schulte, Sep 29 2015; A027907 NAME.
• FORMULA: Conjecture: a(n) = Sum_{j=0..n/2} A027907(n+1-j,2*j+1), n >= 0. - Werner Schulte, Sep 29 2015